Question: A piece of cheese is located at $(12,10)$ in a coordinate plane.  A mouse is at $(4,-2)$ and is running  up  the line $y=-5x+18$.  At the point $(a,b)$ the mouse starts getting farther from the cheese rather than closer to it. What is $a + b$?
Explanation: The  point $(a,b)$ is the foot of the perpendicular from $(12,10)$ to the line $y=-5x+18$.  The perpendicular has slope $\frac{1}{5}$, so its equation is \[
y=10+\frac{1}{5}(x-12)=\frac{1}{5}x+\frac{38}{5}.
\]The $x$-coordinate at the foot of the perpendicular satisfies the equation \[
\frac{1}{5}x+\frac{38}{5}=-5x+18,
\]so $x=2$ and $y=-5\cdot2+18=8$. Thus $(a,b) = (2,8)$, and $a+b = \boxed{10}$.